Final answer:
The implied annual interest rate for a payday loan where $100 is borrowed, a $25 fee is paid upfront, and $112 is to be repaid after one year is 49.33%.
Step-by-step explanation:
To determine the implied interest rate of a payday loan, we first calculate the actual amount of money received today (present value) and the amount that needs to be paid back in the future (future value).
In this case, you borrowed $100 but paid a $25 fee upfront, so the present value is actually $100 - $25 = $75. In one year, you will need to pay back $112.
The interest rate (r) can be calculated using the following formula for simple interest:
Future Value = Present Value (1 + rate × time)
Plugging in the values:
$112 = $75 (1 + rate × 1)
After simplifying:
1 + rate = $112 / $75
rate = ($112 / $75) - 1
rate = 1.4933 - 1
rate = 0.4933, or 49.33%
Therefore, the implied annual interest rate for this payday loan is 49.33%.